Aircraft carriers are more for beating up on nations without substantial resources or power projection once waters have been cleared. Unless you're the US and could stand to lose a few. In a great power conflict, carriers are unlikely to be used in the opening salvos for most nations, since they only have a couple. They're large easy targets and wouldn't be risked.

> There have been multiple publicly known incidents where small submarines surfaced near a carrier without being noticed by any ship or submarine in the carrier group.

I would expect this to be more difficult if the carrier group is on a wartime footing, though.

One of those incidents included a Chinese submarine that surfaced voluntarily 5 miles away. I don't think you can interpret this in any other way than as a demonstration of how they can stalk any carrier near their waters at any time without being noticed.

In 2015 a 30 y/o French submarine "attacked" a US carrier as part of a military exercise where all participants should be on the highest alert. In 2004 a Swedish submarine did the same.